# Augmented Harris Hawks Optimizer with Gradient-Based-Like Optimization: Inverse Design of All-Dielectric Meta-Gratings

## Abstract

**:**

## 1. Introduction

## 2. The Gradient-Based-Like (GBL) Optimization Method

#### 2.1. Initialization

#### 2.2. Evaluation of the Fitness of the Design Parameters

#### 2.3. Best Current Pattern Update

## 3. Harris Hawks Optimizer

#### 3.1. Exploration Phase

#### 3.2. Exploitation Phase ($\left|E\right|<1$)

- 1.
- When $\left|E\right|\ge 0.5$, hawks consider the prey to still have enough energy to escape, and thus, a soft besiege strategy is applied. In this case, locations are updated as follows:$${X}^{(t+1)}=\mathsf{\Delta}{X}^{\left(t\right)}-E|J{X}_{rabbit}^{\left(t\right)}-{X}^{\left(t\right)}|$$
- 2.
- When $\left|E\right|<0.5$, hawks apply a more aggressive strategy, a hard besiege, to capture the prey, as they believe the prey to be too tired to escape. The location update equation for this phase can be written as:$${X}^{(t+1)}={X}_{rabbit}^{\left(t\right)}-E\mathsf{\Delta}{X}^{\left(t\right)}$$
- 3.
- In the exploitation phase, hawks can also perform some progressive rapid dives based on the Levy flight (LF) function. The LF function is defined as:$$LF\left(x\right)=0.01{\displaystyle \frac{u\sigma}{{\left|v\right|}^{1/\beta}}}$$$$\sigma ={\left({\displaystyle \frac{\mathsf{\Gamma}(1+\beta )\times sin\left({\displaystyle \frac{\pi \beta}{2}}\right)}{\mathsf{\Gamma}\left({\displaystyle \frac{1+\beta}{2}}\right)\times \beta \times {2}^{(\beta -1)/2}}}\right)}^{1/\beta}$$$${X}^{(t+1)}=\left\{\begin{array}{c}Y\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}F\left(Y\right)<F\left({X}^{\left(t\right)}\right)\hfill \\ \phantom{\rule{4pt}{0ex}}Z\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}F\left(Z\right)<F\left({X}^{\left(t\right)}\right)\hfill \end{array}\right.$$
- In the case of a soft besiege with progressive rapid dives$$Y={X}_{rabbit}^{\left(t\right)}-E|J{X}_{rabbit}^{\left(t\right)}-{X}^{\left(t\right)}|$$
- In the case of a hard besiege with progressive rapid dives$$Y={X}_{rabbit}^{\left(t\right)}-E|J{X}_{rabbit}^{\left(t\right)}-{X}_{m}^{\left(t\right)}|$$

in both cases $Z=Y+S\times LF\left(D\right)$, where D denotes the dimension of the problem and S is a random vector with size $1\times D$.

## 4. Results and Discussion

## 5. Conclusions and Outlook

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Heidari, A.A.; Mirjalili, S.; Faris, H.; Aljarah, I.; Mafarja, M.; Chen, H. Harris hawks optimization: Algorithm and applications. Future Gener. Comput. Syst.
**2019**, 97, 849–872. [Google Scholar] [CrossRef] - Andrei, N.A. SQP Algorithm for Large-Scale Constrained Optimization: SNOPT. In Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology; Springer Optimization and Its Applications; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Holland, J.H. Genetic algorithms. Sci. Am.
**1992**, 1, 66–73. [Google Scholar] [CrossRef] - Eberhart, R.; Kennedy, J. A new optimizer using particle swarm theory. In Proceedings of the Sixth International Symposium on Micro Machine and Human Science (MHS’95), Nagoya, Japan, 4–6 October 1995; Volume 1, pp. 39–43. [Google Scholar]
- Dorigo, M.; Caro, G.D. Ant colony optimization: A new metaheuristic. In Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406), Washington, DC, USA, 6–9 July 1999; pp. 1470–1477. [Google Scholar]
- Stornand, R.; Price, K. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolves optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The whales optimization algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Senthilnath, J.; Dokania, A.; Kulkarni, S.; Omkar, S.N. Harris hawks optimization for multi-objective optimization in power system engineering. Appl. Soft Comput.
**2020**, 97, 106739. [Google Scholar] - Baskar, S.; Krishnakumar, K. Harris hawks optimization algorithm for segmentation of MRI brain images. Multimed. Tools Appl.
**2021**, 80, 17909–17926. [Google Scholar] - Zhang, H.; Cao, Z. Multimodal optimization and its application in image segmentation based on Harris hawks optimization. Soft Comput.
**2021**, 25, 5019–5033. [Google Scholar] - Sharma, R.; Singh, S.N. An improved Harris hawks optimization with multi-verse optimizer for optimal power flow solution. Int. J. Electr. Power Energy Syst.
**2021**, 128, 106775. [Google Scholar] - Mohajeri, S.; Beheshti, S.M. A novel approach for feature selection based on Harris hawks optimization. J. Ambient. Intell. Humaniz. Comput.
**2020**, 11, 3939–3950. [Google Scholar] - Lim, J.H.; Lee, K.Y. Optimal design of shell-and-tube heat exchanger using Harris hawks optimization. Appl. Therm. Eng.
**2020**, 179, 115656. [Google Scholar] - Qin, N.; Yang, D. Joint base station placement and user association in heterogeneous networks using Harris hawks optimization. Wirel. Netw.
**2020**, 26, 5089–5101. [Google Scholar] - Kaveh, A.; Khayatazad, M. A new optimization method: Ray Optimization. Comput. Struct.
**2012**, 112, 76–82. [Google Scholar] - Song, J.; Li, X.; Zhou, Y.; Li, S. Enhancing the performance of Harris Hawks optimization via persistent-trigonometric-differences mechanism and enhanced energy factor. Soft Comput.
**2021**, 25, 14089–14116. [Google Scholar] - Edee, K. Biomimicry-Gradient-Based Algorithm as Applied to Photonic Devices Design: Inverse Design of Flat Plasmonic Metalenses. Appl. Sci.
**2021**, 11, 5436. [Google Scholar] [CrossRef] - Frandsen, L.H.; Harpøth, A.; Borel, P.I.; Kristensen, M.; Jensen, J.S.; Sigmund, O. Broadband photonic crystal waveguide 60
^{∘}bend obtained utilizing topology optimization. Opt. Express**2004**, 12, 5916–5921. [Google Scholar] [CrossRef] [PubMed] - Borel, P.I.; Harpøth, A.; Frandsen, L.H.; Kristensen, M.; Shi, P.; Jensen, J.S.; Sigmund, O. Topology optimization and fabrication of photonic crystal structures. Opt. Express
**2004**, 12, 1996–2001. [Google Scholar] [CrossRef] - Lu, J.; Vuckovic, J. Nanophotonic computational design. Opt. Express
**2013**, 21, 13351–13367. [Google Scholar] [CrossRef] - Xiao, T.P.; Cifci, O.S.; Bhargava, S.; Chen, H.; Gissibl, T.; Zhou, W.; Giessen, H.; Toussaint, K.C.; Yablonovitch, E.; Braun, P.V. Diffractive spectral-splitting optical element designed by adjoint-based electromagnetic optimization and fabricated by femtosecond 3D direct laser writing. ACS Photonics
**2016**, 3, 886–894. [Google Scholar] [CrossRef] - Sell, D.; Yang, J.; Doshay, S.; Yang, R.; Fan, J.-A. Large-angle, multifunctional metagratings based on freeform multimode geometries. Nano Lett.
**2017**, 17, 3752–3757. [Google Scholar] [CrossRef] - Hughes, T.W.; Minkov, M.; Williamson, I.A.D.; Fan, S. Adjoint Method and Inverse Design for Nonlinear Nanophotonic Devices. ACS Photonics
**2018**, 5, 4781–4787. [Google Scholar] [CrossRef] - Lin, Z.; Groever, B.; Capasso, F.; Rodriguez, A.W.; Lončar, M. Topology-optimized multilayered metaoptics. Phys. Rev. Appl.
**2018**, 9, 044030. [Google Scholar] [CrossRef] - Phan, T.; Sell, D.; Wang, E.W.; Doshay, S.; Edee, K.; Yang, J.; Fan, J.A. High-efficiency, large-area, topology-optimized metasurfaces. Light. Sci. Appl.
**2019**, 8, 48. [Google Scholar] [CrossRef] [PubMed] - Wang, E.W.; Sell, D.; Phan, T.; Fan, J.-A. Robust design of topology-optimized metasurfaces. Opt. Mater. Express
**2019**, 9, 469–482. [Google Scholar] [CrossRef] - Edee, K. Modal method based on subsectional Gegenbauer polynomial expansion for lamellar grating. J. Opt. Soc. Am. A
**2011**, 28, 2006–2013. [Google Scholar] [CrossRef] - Edee, K.; Fenniche, I.; Granet, G.; Guizal, B. Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weighting function, convergence and stability. Prog. Electromagn. Res.
**2013**, 133, 17–35. [Google Scholar] [CrossRef] - Edee, K.; Guizal, B. Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: Complex coordinates implementation. J. Opt. Soc. Am. A
**2013**, 30, 631–639. [Google Scholar] [CrossRef] - Edee, K.; Plumey, J.P. Plumey Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: Application to biperiodic binary grating. J. Opt. Soc. Am. A
**2015**, 31, 402–410. [Google Scholar] [CrossRef] - Edee, K.; Plumey, J.P.; Guizal, B. Unified Numerical Formalism of Modal Methods in Computational Electromagnetics and Latest Advances: Applications in Plasmonics. Adv. Imaging Electron Phys.
**2016**, 197, 45–103. [Google Scholar] - Knop, K. Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves. J. Opt. Soc. Am. A
**1978**, 68, 1206–1210. [Google Scholar] [CrossRef] - Granet, G.; Guizal, B. Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization. J. Opt. Soc. Am. A
**1996**, 13, 1019–1023. [Google Scholar] [CrossRef] - Lalanne, P.; Morris, G.M. Highly improved convergence of the coupled-wave method for TM polarization. J. Opt. Soc. Am. A
**1996**, 13, 779–784. [Google Scholar] [CrossRef] - Li, L. Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A
**1996**, 13, 1870–1876. [Google Scholar] [CrossRef] - Granet, G. Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution. J. Opt. Soc. Am. A
**1999**, 16, 2510–2516. [Google Scholar] [CrossRef] - Jiang, J.; Fan, J.-A. Global Optimization of Dielectric Metasurfaces Using a Physics-Driven Neural Network. Nano Lett.
**2019**, 8, 5366–5372. [Google Scholar] [CrossRef] - Jiang, J.; Lupoiu, R.; Wang, E.W.; Sell, D.; Hugonin, J.P.; Lalanne, P.; Fan, J.A. MetaNet: A new paradigm for data sharing in photonics research. Opt. Express
**2020**, 28, 13670. [Google Scholar] [CrossRef] - Edee, K. Topology optimization of photonics devices: Fluctuation-trend analysis concept; random initial conditions with Gaussian and Durden-Vesecky power density bandlimited spectra. J. Opt. Soc. Am. B
**2020**, 37, 2111–2120. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Device to be designed. The device is made of dielectric nanorods of height ${h}_{1}$, backed on an infinite dielectric substrate. We consider a one-dimensional meta-grating consisting of Si nanorods with a refraction index of $3.6082$, deposited on an SiO${}_{2}$ substrate (refractive index: $1.45$). (

**b**) Example of border-location variables ${x}_{k}$ update. Both widths and spacings of structures are optimized in order to increase the deflected efficiency into the desired diffracted order.

**Figure 4.**Convergence of the transmission efficiency into ${\theta}_{d}={60}^{\circ}$ with respect to the number of iterations $\left(t\right)$ (Y-axis) for different initial randomly generated profiles (X-axis), using the GBL, HHO, and GBL–HHO algorithms. Three values of the parameter ${N}_{p}$ are investigated: ${N}_{p}=7$ (3 nanorods + 4 air gaps), ${N}_{p}=9$ (4 nanorods + 5 air gaps), ${N}_{p}=11$ (5 nanorods + 6 air gaps). Numerical parameters: $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m, deflection angle of ${60}^{\circ}$, polarization TM.

**Figure 5.**Comparison of performances of the GBL optimization (red), HHO (orange) and the proposed hybrid GBL–HHO algorithm (blue), for $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m and the deflection angle of ${60}^{\circ}$. These figures display the efficiency histograms of devices designed using these three methods. Three values of the parameter ${N}_{p}$ are investigated: ${N}_{p}=7$ (3 nanorods + 4 air gaps), ${N}_{p}=9$ (4 nanorods + 5 air gaps), ${N}_{p}=11$ (5 nanorods + 6 air gaps). In (

**a**–

**c**), the GBL optimization is used. (

**d**–

**f**) are devoted to the HHO results. Results obtained from the proposed hybrid GBL–HHO are displayed in (

**g**–

**i**). The minimum size is set to ${e}_{min}=50$ nm.

**Figure 6.**Comparison of performances of GBL optimization (red), HHO (orange) and the proposed hybrid GBL–HHO algorithm (blue), for $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m and a deflection angle of ${40}^{\circ}$. These figures display the efficiency histograms of devices designed using these three methods. Three values of the parameter ${N}_{p}$ are investigated: ${N}_{p}=7$ (3 nanorods + 4 air gaps), ${N}_{p}=9$ (4 nanorods + 5 air gaps), ${N}_{p}=11$ (5 nanorods + 6 air gaps). In (

**a**–

**c**), the GBL optimization is used. (

**d**–

**f**) are devoted to the HHO results. Results obtained from the proposed hybrid GBL–HHO are displayed in (

**g**–

**i**). The minimum size is set to ${e}_{min}=50$ nm.

**Figure 7.**Comparison of performances of GBL optimization (red), HHO (orange) and the proposed hybrid GBL–HHO algorithm (blue), for $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m and a deflection angle of ${80}^{\circ}$. These figures display the efficiency histograms of devices designed using these three methods and for three values of the parameter ${N}_{p}$: ${N}_{p}=7$ (3 nanorods + 4 air gaps), ${N}_{p}=9$ (4 nanorods + 5 air gaps), ${N}_{p}=11$ (5 nanorods + 6 air gaps). In (

**a**–

**c**), the GBL optimization is used. (

**d**–

**f**) are devoted to the HHO results. Results obtained from the proposed hybrid GBL–HHO are displayed in (

**g**–

**i**). The minimum size is set to ${e}_{min}=50$ nm. The highest deflected efficiency in each case is also displayed. For the numerical parameter, the efficiency distributions obtained from the GBL–HHO are narrower than those of GBL and classical HHO.

**Figure 8.**GBL–HHO applied to design a 1D high-transmission deflection meta-grating. Real part and phase of the electric field. Illustration of the quality of the deflection phenomenon supported by highest-transmission devices for three values of the ${\theta}_{d}$: ${40}^{\circ}$ (

**a**,

**b**), ${60}^{\circ}$ (

**c**,

**d**) and ${80}^{\circ}$ (

**e**,

**f**). Numerical parameters: $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m, TM polarization.

**Table 1.**Panel of the efficiencies deflected onto diffracted orders of best devices using the three methods (GBL, HHO, and GBL–HHO) and for three values of ${N}_{p}\in \{7,9,11\}$. The best optimized device obtained from both the GBL optimizer and GBL–HHO have higher maximum efficiency values compared to those from the classical HHO. Numerical parameters: $\lambda =0.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m, ${h}_{1}=325$ nm.

Method | GBL | HHO | GBL–HHO | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{N}}_{\mathit{p}}$ | 7 | 7 | 11 | 7 | 7 | 11 | 7 | 7 | 11 |

${40}^{\circ}$ | $85.4\%$ | $95.4\%$ | $95.1\%$ | $85.2\%$ | $93.8\%$ | $92.1\%$ | $85.4\%$ | $95.2\%$ | $95.6\%$ |

${60}^{\circ}$ | $92.8\%$ | $98.0\%$ | $98.4\%$ | $94.0\%$ | $93.8\%$ | $75.4\%$ | $93.8\%$ | $98.0\%$ | $98.0\%$ |

${80}^{\circ}$ | $79.4\%$ | $93.5\%$ | $88.0\%$ | $85.1\%$ | $92.8\%$ | $82.6\%$ | $87.0\%$ | $93.6\%$ | $91.4\%$ |

Method | Adjoint-Based TO | GLOnets Optimization |
---|---|---|

${40}^{\circ}$ | $88\%$ | $87\%$ |

${60}^{\circ}$ | $81\%$ | $94\%$ |

${80}^{\circ}$ | $72\%$ | $89\%$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Edee, K.
Augmented Harris Hawks Optimizer with Gradient-Based-Like Optimization: Inverse Design of All-Dielectric Meta-Gratings. *Biomimetics* **2023**, *8*, 179.
https://doi.org/10.3390/biomimetics8020179

**AMA Style**

Edee K.
Augmented Harris Hawks Optimizer with Gradient-Based-Like Optimization: Inverse Design of All-Dielectric Meta-Gratings. *Biomimetics*. 2023; 8(2):179.
https://doi.org/10.3390/biomimetics8020179

**Chicago/Turabian Style**

Edee, Kofi.
2023. "Augmented Harris Hawks Optimizer with Gradient-Based-Like Optimization: Inverse Design of All-Dielectric Meta-Gratings" *Biomimetics* 8, no. 2: 179.
https://doi.org/10.3390/biomimetics8020179